Help with finding the length of a cosine wave

[Flax]

Hello,

I am new to this forum and I apologise if I have placed this question in the wrong place and will move it if needed. I have a formula that is written as f(x) = A*cos((2*pi*x)/T) , A being Amplitude of the Cosine wave and T being the period. when I put this into a sketching program the function preforms how I presume it should. The Question I have is that I would like to calculate the length of the wave between 2 points and I presumed this would use the integral method, how ever it's been a very long time since I've done any sort of math and it didn't work how I thought it would.

I inputted ∫ 10 , 0 10*cos((2*pi*x)/10)

as an example and got an answer of

100*cos((x*pi)/5) which is not what I was expecting.

If anyone would be able to help me in the right direction it would be really appreciated, I am sure I am just missing some information along the way.

Thankyou

 

[HallsofIvy]

I don't know how you got a "cosine" again in the integral. The integral \(\displaystyle \int cos(ax)dx\) is \(\displaystyle \frac{1}{a}sin(ax)\). However, the arclength of y= f(x) from x= p to x= q is not \(\displaystyle \int_p^q f(x)dx\). That is the are under the curve. The arclength is, rather, \(\displaystyle \int_p^q \sqrt{1+ f'(x)^2}dx\). With \(\displaystyle f(x)= 10 cos(\pi x/5)\), \(\displaystyle f'= 2\pi sin(\pi x/5)\) so that \(\displaystyle f'^2= 4\pi^2 sin^2(\pi x/5)\), \(\displaystyle 1+ f'^2= 1+ 4\pi sin^2(\pi x/5)\) and \(\displaystyle \sqrt{1+ f'^2}= \sqrt{1+ 4\pi sin^2(\pi x/5)}\). That is what you want to integrate. I don't be believe that has an elementary anti-derivative. It looks like an "elliptic integral" to me.

 

[Flax]

Thank you for your response, I will be honest and say a lot of that went over my head upon first read, so I will spend a bit of time breaking each part down, but yes I realised from the time of making this post and it being published that I can't just put the initial formula into the Integral function. From what I'm reading it seems I wasn't correct with that line of thought either and this is seeming less and less like highschool math haha, I appreciate the help I will read over it and return if I am unable to make heads and tails of your answer.
[FONT=MathJax_Math-italic][/FONT]